September 09, 2009
63

# Introduction to Mesoscale Methods

Lecture at the Understanding Molecular Simulations school at the Jawaharlal Nehru Center for Advanced Scientific Research.

September 09, 2009

## Transcript

1. Mesoscale Simulation Methods
The Institute of Mathematical Sciences
Chennai

2. Outline
• What is mesoscale ?
• Mesoscale statics and dynamics through coarse-graining.
• Coarse-grained equations for a binary ﬂuid-ﬂuid mixture.
• Numerical methods of solution.
• Example simulations.

3. Numbers and methods
1
2
3
4
5
6
7
0
Ab initio methods
Atomistic methods
Continuum methods
DFT
CPMD
MD
LD
BD
DPD
CFD
LB
Number of atoms (log) Simulation method Example of method

4. How lengths scale with numbers
!! Try this for water
Molar mass = 18 gm
Molar volume = 18 mL
N = 1023
L 107˚
A
L N 1
3
N = 1
L 1˚
A
!! Are other scalings possible ? How does length grow with
size for a polymer ?
Throughout, (!!) indicate exercises/derivations/points to ponder.

5. Length and time scales
pico
milli
micro
nano
nano micro milli
L
T
H = E
m ¨
R = ⇥U
˙
R = U
˙
c = D 2c
!! Are there systems with millimeter L but picosecond T ?

6. How times scale with masses and lengths

x + kx = 0
2
0
= k
m
0
m
k
¨
u = c2 2u
0
= ±cq
⇥0 c
˙
u = D 2u
i 0
= Dq2
⇥0
2
D
Harmonic Oscillator Wave Equation Diffusion Equation

7. Thermal ﬂuctuations
P(x |x) =
1

2 D⇥
exp
(x x)2
2D⇥

Brownian motion : Einstein (1905)
D = kBT
6⇥ a
temperature
size
Stokes-Einstein-Sutherland Relation
Q. Why does a smaller particle have a higher
diffusion coefﬁcient ?
A. The ‘root N’ effect! The central limit theorem
helps us understand this.
!! Is it (central limit) theorem or
central (limit theorem) ?

8. Langevin theory
inertia
damping reversible force
ﬂuctuation
deterministic stochastic
mean behaviour ‘root N’ ﬂuctuations
regression to the mean
equilibrium ﬂuctuations
v2⇥ = kBT
F = 0 Free Brownian particle

v + v F(x) = ⇥(t)

9. Mesoscale regime
1
2
3
4
5
6
7
0
Ab initio
Atomistic
Continuum
Number of atoms Simulation
Coarse grained length scales.
Coarse grained time scales.
Retain thermal ﬂuctuations.
Mesoscale methods
}
Examples
Brownian dynamics.
Dissipative particle dynamics.
Time-dependent Ginzburg-Landau.

10. Coarse-graining in degrees of freedom
What is the idea ?
x1
x2
} P(x1, x2
)
RV y = x1
+ x2
“Coarse-grained” sum
P(y)
P(y) = dx1dx2
(y x1 x2
)P(x1, x2
)
What is the distribution ?
contains less information about the system than
P(y) P(x1, x2
)
Adequate if we are only interested in the sum variable.

11. General ‘coarse-graining’ formula
P(x1, x2, . . . , xN
) y = f(x1, x2, . . . , xN
)
Microstate probability Mesoscale variable/order parameter
P(y) =
⇥ N
i=1
dxi
[y f(x1, . . . , xN
)]P(x1, . . . , xN
)
Mesostate probability

12. Coarse-grained Landau-Ginzburg functional
Microscopic Hamiltonian Gibbs distribution
P(q) =
exp[ H(q)]
Z
H(q)
= f(q)
Order parameter Deﬁnition of Landau functional
P(⇤) =
⇥ N
i=1
dqi⇥[⇤ f(q)]
exp[ H(q)
Z

exp[ L]
Z
P(⇥) ⇥
exp[ L]
Z

13. Explicit DOF coarse-graining ...
H(s1, s2, . . . , sN
) = J
ij⇥
sisj
Order parameter.
Block spin.
Coarse-grained
magnetization.
=
1
N
i
si

14. Explicit DOF coarse-graining ...
H(s1, s2, . . . , sN
) = J
ij⇥
sisj
Order parameter.
Block spin.
Coarse-grained
magnetization.
=
1
N
i
si
P(⇥ ) ⇥
exp[ L]
Z⇥
=?
K. Binder, Z. Phys B, 43, 119 (1981)

15. Explicit DOF coarse-graining ...
H(s1, s2, . . . , sN
) = J
ij⇥
sisj
Order parameter.
Block spin.
Coarse-grained
magnetization.
=
1
N
i
si
P(⇥ ) ⇥
exp[ L]
Z⇥
=?
K. Binder, Z. Phys B, 43, 119 (1981)

16. Explicit DOF coarse-graining ...
H(s1, s2, . . . , sN
) = J
ij⇥
sisj
Order parameter.
Block spin.
Coarse-grained
magnetization.
=
1
N
i
si
P(⇥ ) ⇥
exp[ L]
Z⇥
=?
K. Binder, Z. Phys B, 43, 119 (1981)
Explicit DOF needs this!

17. Or ansatz based on symmetry.
L = A
2
2 + B
4
4 + K
2
(⇥ )2
P(⇥) ⇥ exp[ (A
2 ⇥2 + B
4 ⇥4)]
P(s) ⇥ a+
(s 1) + a (s + 1) -J +J
local part non-local part
K( )2
All equal-time equilibrium properties are determined by this functional.
!! Why is this called a “free energy” ? Is it a thermodynamic free energy ?

18. What about dynamics ? Temporal coarse-graining
D = kBT
6⇥ a
Why is this independent of
particle mass ?
⇥d
= m Inertial time scale. Time the
velocity ‘remembers’ its
initial condition.
d
˙
x = v
˙
x = ⇥(t)
Overdamped equations of
motion. Valid when
d

v + v = ⇤(t)
= 6⌅⇥a

19. Overdamped stochastic equations
damping reversible force
ﬂuctuation
deterministic stochastic
mean behaviour ‘root N’ ﬂuctuations
˙
x = ⇥U + ⇥(t)

20. Overdamped coarse-grained equations of motion
˙
⇤(r) =
L

+ ⇥(r, t)
˙
⇥(r) = [A⇥ + B⇥3 K⇥2⇥] + (r, t)
damping reversible force
ﬂuctuation
Model A equation
⇥⇥(r, t)⇥(r , t )⇤ = 2kBT (r r ) (t t )
Fluctuation Dissipation
Relation.
!! Zero-mean
Gaussian noise
(r, t)⇥ = 0
“Stochastic partial differential equations with additive Gaussian noise”

21. Conserved order parameters
Ising spin Lattice gas
Ising spins are usually non-conserved
d
dt
(r, t) = 0
Lattice gas models are conserved
d
dt
(r, t) = 0

22. Conserved coarse-grained dynamics
damping reversible force
ﬂuctuation
Model B equation
Fluctuation Dissipation
Relation.
!! Zero-mean vector
Gaussian noise
(r, t)⇥ = 0
˙
⇤(r) = ⇤2
L

+ ⇤ · ⇥(r, t)
˙
⇥(r) = ⇤2[A⇥ + B⇥3 K⇤2⇥] + ⇤ · (r, t)
!! Derive this. Use the local conservation law
˙ + ⇥ · j = 0
⇥⇥i
(r, t)⇥j
(r , t )⇤ = 2kBT ij
(r r ) (t t )

23. Numerical solution
Stochastic PDE
Stochastic ODE
Stochastic realisation
Spatial discretisation
Temporal integration
⇤t⇥ = D⇤2
x
⇥ +
⇥x
=
(x + h) (x)
h
⇤t⇥(i) = DL2
ij
⇥(j) + (i)
⇥(i, t + t) ⇥(i, t) =
t+
t
DL2
ij
⇥(j) + (i)
!! Obtain the explicit form of the L matrix for a central difference Laplacian

24. Summary of TDGL mesoscale methods
Microscopic Hamiltonian
Ginzburg-Landau Functional
Langevin equations
Explicit Coarse Graining
Overamped approximation
Numerical solution

25. Conclusion
• Mesoscale methods are appropriate at length scales intermediate between the molecular
and continuum scales.
• The continuum description must be supplemented by ﬂuctuation terms. At the mesoscale,
the number of particles is not so large that ﬂuctuations can be neglected.
• Lengths and times must be coarse-grained. Intelligent coarse-graining improves
computational eﬃciency, often by orders of magnitude, compared to direct MD.
• Mesoscale methods can be particle-based, for instance dissipative particle dynamics and
Brownian dynamics, or ﬁeld-based like time-dependent Ginzburg-Landau theory and
ﬂuctuating hydrodynamics.
• TDGL equations can be solved very eﬃciently on computers using matrix formulations.